Exercise 9.3.6: Suppose U1 and U2 are two open sets in R n with U1 ?U2 nonempty and path connected. Suppose there exists an f1 : U1 ? R and f2 : U2 ? R, both twice continuously differentiable such that d f1 = d f2 on U1 ?U2. Then there exists a twice differentiable function F : U1 ?U2 ? R such that dF = d f1 on U1 and dF = d f2 on U2

Exercise 9.3.6: Suppose U1 and U2 are two open sets in R" with UinU2 nonempty and path connected. Suppose there exists an fi : U1 - R and f2: U2 - R, both twice continuously differentiable such that dfi = df2 on UnU2. Then there exists a twice differentiable function F : U UU2 -> R such that dF = dfi on U1 and dF = df2 on U2